Somewhere injective vs. multiply covered

This week I’m at the AIM workshop on transversality in contact homology, and I must say I am impressed. Given several choices of topics for possible discussion, for the first time this week I have witnessed an overwhelming plurality of the people in a room shunning various “soft” topics involving applications and computations, because they prefer instead to hear about things like obstruction bundle gluing. This is my kind of workshop.

Anyway, I’ve been covering various foundational folk theorems on this blog lately, and this post will be another in that genre, prompted in this case by a specific question that was asked at the workshop. The question was whether a proof of the standard dichotomy between somewhere injective and multiply covered holomorphic curves has been written down in the setting of SFT. While it is easy to find readable proofs of the corresponding result for closed holomorphic curves (e.g. see the two proofs of Proposition 2.5.1 in McDuff-Salamon), there is some reason to be suspicious of casual statements that the result generalizes with no trouble, e.g. it has been well known for a long time that the corresponding result for holomorphic disks with totally real boundary is not true. Nonetheless, the result does hold for asymptotically cylindrical punctured curves without boundary, and one can prove it in much the same way as in the closed case, one only needs an extra lemma about asymptotic behavior. The original proof for the special case of holomorphic planes appeared in the Appendix of HWZ “Properties II”, and I have occasionally heard it claimed that the HWZ proof can be adapted for the general case, though I have never sat down to verify this claim. The proof I am going to explain here is modeled on the first of the two proofs for the closed case in the McDuff-Salamon book, which also appears as Theorem 2.120 in my holomorphic curve notes.

Here is the statement.

Theorem. Assume $(W,J)$ is an almost complex manifold with cylindrical ends adapted to stable Hamiltonian structures, and $u : (\dot{\Sigma},j) \to (W,J)$ is a nonconstant asymptotically cylindrical $J$ -holomorphic curve asymptotic to nondegenerate or Morse-Bott Reeb orbits. Let $(\Sigma,j)$ denote the closed Riemann surface from which $(\dot{\Sigma},j)$ is obtained by deleting finitely many points. Then there exists a factorization $u = v \circ \varphi$ , where

$v : (\dot{\Sigma}',j') \to (W,J)$ is an asymptotically cylindrical $J$ -holomorphic curve that is embedded outside a finite set of critical points and self-intersections, and
$\varphi : (\Sigma,j) \to (\Sigma',j')$ is a holomorphic map of positive degree, where $(\dot{\Sigma}',j')$ is obtained from the closed Riemann surface $(\Sigma',j')$ by deleting finitely many points.

The main idea is to construct $\dot{\Sigma}'$ (minus some extra punctures) explicitly as the image of $u$ after removing finitely many singular points, so that we can take $v$ to be the inclusion $\dot{\Sigma}' \hookrightarrow W$ ; the map $\varphi : \dot{\Sigma} \to \dot{\Sigma}'$ is then uniquely determined and extends holomorphically over all punctures due to removal of singularities. In order to carry out this program, we need some information on what the image of $u$ can look like near each of its singularities. These come in three types, each type corresponding to one of the lemmas below. The first two should seem immediately plausible if your intuition comes from complex analysis.

Intersection lemma. Suppose $u : (\Sigma,j) \to (W,J)$ and $v : (\Sigma',j') \to (W,J)$ are two nonconstant pseudoholomorphic curves with an intersection $u(z) = v(z')$ . Then there exist neighborhoods $z \in {\mathcal U} \subset \Sigma$ and $z' \in {\mathcal U}' \subset \Sigma'$ such that

either $u({\mathcal U}) = v({\mathcal U}')$ or $u({\mathcal U} \setminus \{z\}) \cap v({\mathcal U}' \setminus \{z'\}) = \emptyset$ .

Branching lemma. Suppose $u : (\Sigma,j) \to (W,J)$ is a nonconstant pseudoholomorphic curve and $z_0 \in \Sigma$ is a critical point of $u$ . Then a neighborhood ${\mathcal U} \subset \Sigma$ of $z_0$ can be biholomorphically identified with the unit disk ${\mathbb D} \subset {\mathbb C}$ such that

$u(z) = v(z^k)$ for $z \in {\mathbb D} = {\mathcal U}$ ,

where $k \in {\mathbb N}$ , and $v : {\mathbb D} \to W$ is an injective $J$ -holomorphic map with no critical points except possibly at the origin.

These two local results follow from a well-known formula of Micallef and White describing the local behavior of $J$ -holomorphic curves near critical points and their intersections. The proof of that theorem is analytically quite involved, but one can also use an easier “approximate” version, which is proved in Section 2.13 of my holomorphic curve notes. The main idea in either case is that since every almost complex structure is locally $C^\infty$ -close to one that is integrable, the local behavior of $J$ -holomorphic curves should also match the integrable case.

We now have all the ingredients needed to prove the theorem for closed $J$ -holomorphic curves. For curves with punctures, we need one more ingredient.

Asymptotics lemma. Assume $u : (\dot{\Sigma},j) \to (W,J)$ is asymptotically cylindrical and $z_0 \in \Sigma$ is a puncture at which the asymptotic orbit is nondegenerate or Morse-Bott. Then a punctured neighborhood $\dot{\mathcal U}$ of $z_0$ in $\dot{\Sigma}$ can be identified biholomorphically with the punctured unit disk $\dot{\mathbb D} = {\mathbb D} \setminus \{0\} \subset {\mathbb C}$ such that

$u(z) = v(z^k)$ for $z \in \dot{\mathbb D} = \dot{\mathcal U}$ ,

where $k \in {\mathbb N}$ and $v : \dot{\mathbb D} \to W$ is an embedded asymptotically cylindrical curve.

Moreover, if $w : (\dot{\Sigma}',j') \to (W,J)$ is another asymptotically cylindrical curve with a puncture $z_0' \in \Sigma'$ , then the images of $u$ near $z_0$ and $w$ near $z_0'$ are either identical or disjoint.

Note that the second statement in this result is obvious if $u$ and $w$ are asymptotic to different orbits at the two punctures under consideration, but importantly, it’s still true if both are asymptotic to (covers of) the same orbit. This lemma follows from a set of relative asymptotic formulas proved in a paper of Siefring, where he uses them to establish an asymptotic version of the Micallef-White formula. These results are the main technical engine behind Siefring’s intersection theory of punctured holomorphic curves, but they are of much wider interest than that, e.g. while intersection theory only makes sense in dimension 4, the asymptotics lemma is valid in all dimensions. I do not know whether the Morse-Bott condition really needs to be assumed in the lemma — I would guess that it can be weakened but not completely removed, as one always needs some such condition to make sure that punctured holomorphic curves behave reasonably near infinity. (Hofer had no such assumption in his 1993 Weinstein conjecture paper, but to be honest, the asymptotic behavior of the curves in that paper cannot really be called reasonable.)

Proof of the theorem. Let $\text{Crit}(u) = \{ z \in \dot{\Sigma}\ |\ du(z) = 0 \}$ denote the set of critical points, and define $\Delta \subset \dot{\Sigma}$ to be the set of all points $z \in \dot{\Sigma}$ such that there exists $z' \in \dot{\Sigma}$ and neighborhoods $z \in {\mathcal U} \subset \dot{\Sigma}$ and $z' \in {\mathcal U}' \subset \dot{\Sigma}$ with $u(z) = u(z')$ but

$u({\mathcal U} \setminus \{z\}) \cap u({\mathcal U}' \setminus \{z'\}) = \emptyset.$

The three lemmas quoted above imply that both of these sets are discrete and they have no accumulation points near the punctures. Both are therefore finite, and the set

$\ddot{\Sigma}' = u(\dot{\Sigma} \setminus (\text{Crit}(u) \cup \Delta)) \subset W$

is then a smooth submanifold of $W$ with $J$ -invariant tangent spaces, so it inherits a natural complex structure $j'$ for which the inclusion $(\ddot{\Sigma}',j') \hookrightarrow (W,J)$ is pseudoholomorphic. We shall now construct a new Riemann surface $(\dot{\Sigma}',j')$ from which $(\ddot{\Sigma}',j')$ is obtained by removing a finite set of points. Let

$\widehat{\Delta} = (\text{Crit}(u) \cup \Delta) / \sim$ ,

where two points in $\text{Crit}(u) \cup \Delta$ are defined to be equivalent whenever they have neighborhoods in $\dot{\Sigma}$ with identical images under $u$ . Then for each $[z] \in \widehat{\Delta}$ , the branching lemma provides an injective $J$ -holomorphic map $u_{[z]}$ from the unit disk ${\mathbb D}$ onto the image of a neighborhood of $z$ under $u$ . We define $(\dot{\Sigma}',j')$ by

$\dot{\Sigma}' = \ddot{\Sigma}' \cup_\Phi \left( \bigsqcup_{[z] \in \widehat{\Delta}} {\mathbb D} \right)$ ,

where the gluing map $\Phi$ is the disjoint union of the maps $u_{[z]} : {\mathbb D} \setminus \{0\} \to \ddot{\Sigma}'$ for each $[z] \in \widehat{\Delta}$ ; since this map is holomorphic, the complex structure $j'$ extends from $\ddot{\Sigma}'$ to $\dot{\Sigma}'$ . Combining the maps $u_{[z]} : {\mathbb D} \to W$ with the inclusion $\ddot{\Sigma}' \hookrightarrow W$ now defines a pseudoholomorphic map $v : (\dot{\Sigma}',j') \to (W,J)$ which restricts to $\ddot{\Sigma}'$ as an embedding and otherwise has at most finitely many critical points and double points. Moreover, the restriction of $u$ to $\dot{\Sigma} \setminus (\text{Crit}(u) \cup \Delta)$ defines a holomorphic map to $(\ddot{\Sigma}',j')$ which extends by removal of singularities to a proper holomorphic map $\varphi : (\dot{\Sigma},j) \to (\dot{\Sigma}',j')$ such that $u = v \circ \varphi$ . It remains only to observe that by the first statement in the asymptotics lemma, the complements of certain compact subsets in $(\dot{\Sigma},j)$ and $(\dot{\Sigma}',j')$ can each be identified biholomorphically with punctured disks $\dot{\mathbb D}$ on which $\varphi(z) = z^k$ for various $k \in {\mathbb N}$ . One can therefore glue disks to $\dot{\Sigma}'$ so that it becomes the complement of a finite set of punctures in some closed Riemann surface $(\Sigma',j')$ , and $\varphi$ extends to a nonconstant holomorphic map $(\Sigma,j) \to (\Sigma',j')$ . $\Box$